I have seen lately many people who tell that Jon will make his move to claim the throne or his birthright based on the fact that his birthfather is Rhaegar but I disagree. I believe that others will try to put Jon on the throne once his true identity become a common knowledge (something like Wyman tries to do with Rickon)but Jon will never actively try to do it by himself because as he said

Lord Eddard Stark is my father. I will not forget him, no matter how many swords they give me.

I refuse to believe that Jon will attempt to sit the throne. His story is in the North, and Jon embodies the North(south of the wall) more than any other Stark. I believe that his parentage will give him a sense of peace, and allow him to realize that he was not conceived in disgrace, nor was he just an accidental bastard child. Lord Commander of a re-organized Night's Watch, or being crowned the King in the North would be acceptable paths for Jon.

Do you remember the scene where Jon Snow was remembering when playing at Swords with Robb how they would call out great warriors names, and Jon remembered the time he called out "Lord of Winterfell", and then reflecting he felt bad about it? He also felt bad about dreaming for a day when he could prove (likely with feat of arms) he was a true Stark and his father would bestow Ice on him? He clearly yearned to be accepted as a Stark, yet he turn down the chance to rule Winterfell when it came. He didn't do it because he didn't want to displace his brothers (who he thought were all dead), but because he feels his place is at the wall. After all that I don't think he would take any crown, or leave the Night's Watch for any reason.

That being said, IF R+L=J is true, I think that the knowledge that his parents were in love and not just that one time Ned forgot his honor will be what has the most impact on him. Plus, he's wanted to know his mother for so long, and didn't even know if she was alive, knowing she passed long ago might help him as well because then he knows why his mother didn't have a relationship with him growing up.

If the war with the Others becomes impossible to fight because of scheming in the South I can see Jon needing to take the throne in the future. We saw with the pink letter that Jon will not just sit at the Wall waiting for people to take shots at him. If Cersei and other people playing the game of thrones are going to continue to send assassins and trouble makers to the Wall then Jon could use R+L=J as a reason to take power from them.

Unless GRRM goes character-assassination on Jon, he won't claim Winterfell as long as Sansa, Arya, Bran and Rickon live and he knows about it. Much less the Iron Throne. After all, he is his father's son.

I also doubt that Jon will seek to sit the IT. Whatever is to become of his character, if he won't remain in the NW he'll leave Winterfell for any remaining Stark. In any case, I don't think he'll want to break his vows again:"...I shall wear no crowns and win no glory..."

I think I have to make it clear: I don’t believe that he will actively try to claim the Iron(or the Northern) Throne but he will accept the Throne when other functions (GNC participants maybe) will give it to him.

Unless GRRM completely changes Jons character, he'll never seek to claim the throne. Rhaegar may have impregnated his mother but Ned is his father and Jon is a Northman.

ITA with this. He will still be Ned Stark's son, regardless of dna. The R+L=J information may be a personal balm to him, but in the long run, he is as much Stark as Leanna, just as Ned raised him to be. Luckily, his 'wild wolf' side isn't out of control, like his mother's may have been.

If ASOIAF was an average fantasy, Jon would probably use his heritage as a point of pride (after angsting about it for half a chapter) and be recognized by everyone as the true prince who was promised. However, Jon was raised to consider the R+L story as one of abduction and rape. If he does find out about his parents, it won't be something he'll be proud of. I doubt very much that GRRM will have someone like Howland Reed pop in and say "hey, you're really a prince and your parents actually loved each other!" He grew up as a bastard, but at least of a man he loved and respected. I'm pretty sure he would prefer that any day over being a bastard born of rape to a man he never knew and would have grown up thinking pretty poorly of.